Q63PE - College Physics (Urone)

Question

Question: What is the dew point (the temperature at which 100% relative humidity would occur) on a day when relative humidity is 39.0% at a temperature of 20.0°C?

Step-by-Step Solution

Answer

Answer

The temperature is 5°C at the dew point.

Step 1: Define relative humidity

The relative humidity is the ratio of vapor density to the saturation vapor density expressed in percentage.

$\mathbf{\text{Percent }}\mathbf{ }\mathbf{\text{Relative Humidity}}\mathbf{ }\mathbf{\text{ = }}\mathbf{ }\frac{\mathbf{\text{Vapor }}\mathbf{ }\mathbf{\text{Density}}}{\mathbf{\text{Saturation Vapor }}\mathbf{ }\mathbf{\text{Density}}}\mathbf{\times }\mathbf{100}$…………….(1)

Here, vapor density is constant and saturation vapor density decreases with the temperature.

Step 2: Find saturation vapor density at 100%

Saturation vapor density corresponding to 20.0°C is 17.2 g/m³. Relative humidity at 20.0°C is given as 39.0%. Substitute these values in equation (1) for both values of relative humidity.

$$\begin{gathered} 39.0\%=\frac{vapor density}{17.2 g/m^3}\times 100 \\ 100\%=\frac{vapor density}{Saturation Vapor Density \text{at} \text{100\% } \text{relative} \text{humidity}} \end{gathered}$$

Dividing these two equations gives;

$$\begin{gathered} \frac{{\displaystyle Saturation Vapor Density \text{at} \text{100\% } \text{relative} \text{humidity}}}{17.2g/m^3}=\frac{39.0\%}{100\%} \\ Saturation Vapor Density \text{at} \text{100\% } \text{relative} \text{humidity}=\frac{39.0\%}{100\%}\times 17.2g/m^3 \\ =6.71g/m^3 \end{gathered}$$

The temperature corresponding to a saturation vapor density of 6.80 g/m³ is 5°C. Since the saturation vapor density decreases with temperature, the temperature that corresponds to a saturation vapor density of 6.71 g/m³ will be a little less than 5°C.

So, at 100% relative humidity, the temperature is approximately equal to 5°C.